Reconstructions Of Heat Sources For 1-dimensional Heat Equation

The inhomogeneous term in the heat conduction equation is important for heat conduction process,which can be considered as one of the heat sources of the heat pro-cess.If all the source terms are known,the boundary state and the initial temperature distribution of the heat conduction process are given,the solution of the conduction sys-tem can be obtained by solving the direct problem for the parabolic equation,which is well-posed.However,for many practical problems,the inhomogeneous term of the heat equation,some parameters in the equation as well as the definition domain of the conduc-tion process may be unknown.In these cases,we have to determine the heat conduction process by firstly identify the unknown ingredients of the conduction system from some measurable data of the temperature fields and then determine the heat field.Such kinds of problem are called inverse problems for partial differential equations.In many natural science and engineering fields,one often encounters the unknown source term of the heat equation,which is distributed inside the entire medium and is difficult to measure directly.An implementable approach is to use the temperature field that can be measured direct-ly as an additional data,and consequently the heat conduction model with such extra data is used to identify the internal heat source distribution,and then the temperature distribution of the entire medium.In this paper,we consider the following heat conduction model for the temperature field u(x,t)where QT= {(x,t)|0<x<l,0<t<T},Ω = {x|0≤x≤l}.The unknown source term f has the following three forms:time dependent,space-wise dependent,time and space dependent,while the function g(x,t)is known.The inverse problem is to reconstruct the unknown source term f(t),f(x),f(x,t)respectively,from one of the following nonlocal observation data:In contrast to the inverse problem of given point by point measurements for additional data,the integral form of the input data is actually a weighted average of the temperature field,more in line with the engineering practice.This paper consists of the following five parts.We firstly give an overview on the engineering background and existing works on the inverse problems for heat conduction process in chapter 1,Then the purpose and main research contents of this paper are stated.In chapter 2,we introduce some necessary knowledge applied for inverse problems,which provide the theoretical basis for the reconstruction of the source terms for heat conduction equation in our work.In chapter 3,we consider the reconstructions of source terms in three different forms respectively.For the first two forms of source terms of single variable which depends either on t or x,the existence and uniqueness,conditional stability,convergence of the approximate regularizing solution of the inverse problems are proved.We also propose some choice strategy for the regularizing parameters of the regularizing solution.For the source term which depends both on time variable and space variable,the solution of the inverse problem is not unique.The existence and uniqueness of the minimizers of the cost functional,with a rigorous analysis on the convergence property of the functional,is proved.Also,the influence of the number of observation data on performance of the solution to the inverse problems is analyzed.For the first two forms of source terms of single variable which depends either on t or x can be be regarded as the special form of f(x,t),and our numerical reconstruction method is the optimization problem based on Tikhonov regularizing functional,so in chapter 4,we establish an iterative algorithm for solving Tikhonov regularizing functional of f(x,t).For the numerical realizations,the problems for the unknown sources are reformulated as a least squares problem in coupling with a Tikhonov regularizing term.It is proved that the Tikhonov functional Jγ(f)is Frechet differentiable and the formula for the gradient of the cost functional is derived via an adjoint problem,and the conjugate gradient method(CGM)is applied to construct the numerical solutions.The first two unknown function forms of a single variable are the same as here.Finally,we present some numerical examples to show the efficiency of the proposed algorithms.We conclude our thesis by chapter 5,where we summarize our researches and give some comments for further works.